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Homework Help: Prove that similar matrices have the same rank

  1. Feb 19, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that similar matrices have the same rank.

    2. Relevant equations

    3. The attempt at a solution

    Similar matrices are related via: B = P-1AP, where B, A and P are nxn matrices..
    since P is invertible, it rank(P) = n, and so since the main diagonal of P all > 0, multiplying by P will not change the rank of A, so rank B = rank A.

    Is that seem right?
  2. jcsd
  3. Feb 19, 2010 #2
    If you can show that rank(P-1AP) is less than or equal to rank(A), then you are done since matrix similarity is symmetric (if A is similar to B, then B is similar to A).
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